# Rotation matrix proof

1. Nov 3, 2015

### whatisreality

1. The problem statement, all variables and given/known data
Let A∈M2x2(ℝ) such that ATA = I and det(A) = -1. Prove that for ANY such matrix there exists an angle θ such that

A = $\left( \begin{array}{cc} cos(\theta) & sin(\theta)\\ sin(\theta) & -cos(\theta)\\ \end{array} \right)$

It is not sufficient to show that this matrix satisfies the specified relations.
2. Relevant equations

3. The attempt at a solution
Where do I start with this?! I'm supposed to get from ATA = I to the rotation matrix! I have looked at several proofs online. But they seem to either use eigenvalues and vectors (which we haven't done, so can't use them!) or don't mention the properties I've been given.

I know a few things that might be useful.
$(A^T)^{-1} = (A^{-1})^T$
And $det(A^T) = det(A)$
Also, I know that the matrix A is orthogonal.

Don't know how to start!

2. Nov 3, 2015

### Ray Vickson

$$A = \pmatrix{a &b \\ c& d}$$

Evaluate $P_1 = A A^T$ and $P_2 = A^T A$. You need both $P_1 = I$ and $P_2 =I$, and those will give you several equations that the entries $a,b,c,d$ must satisfy. You also need $\det(A) = 1$, giving you $ad - bc = 1$.

3. Nov 3, 2015

### PeroK

If you have $a^2 + b^2 = 1$, can you show that there exists $\theta$ such that $a = cos\theta$ and $b = sin\theta$?

4. Nov 7, 2015

### whatisreality

Got there. Thank you!